Melting of Two-Dimensional Tunable-Diameter Colloidal Crystals

PHYSICAL REVIEW E 77, 041406 ͑2008͒

Melting of two-dimensional tunable-diameter colloidal crystals

Y. Han,* N. Y. Ha,† A. M. Alsayed, and A. G. Yodh Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6396, USA ͑Received 12 February 2007; revised manuscript received 24 November 2007; published 18 April 2008͒ Melting of two-dimensional colloidal crystals is studied by video microscopy. The samples were composed of microgel spheres whose diameters could be temperature tuned, and whose pair potentials were measured to be short ranged and repulsive. We observed two-step melting from the crystal to a hexatic phase and from the hexatic to the liquid phase as a function of the temperature-tunable volume fraction. The translational and orientational susceptibilities enabled us to deﬁnitively determine the phase transition points, avoiding ambiguities inherent in other analyses and resolving a “dislocation precursor stage” in the solid phase that some of the traditional analyses may incorrectly associate with the hexatic phase. A prefreezing stage of the liquid with ordered patches was also found.

DOI: 10.1103/PhysRevE.77.041406 PACS number͑s͒: 82.70.Ϫy, 64.60.Cn, 64.70.DϪ

I. INTRODUCTION to follow the spatiotemporal evolution of the same sample area through the entire sequence of transitions. This feature ͑ ͒ Two-dimensional 2D melting is a classic problem in is attractive and was not realized in previous colloidal melt- ͓ ͔ condensed matter physics 1 , and, over the years, theories ing experiments which employed samples composed of attempting to understand 2D melting have emphasized topo- charged spheres in the wedge geometry with density gradi- ͓ ͔ ͓ ͔ logical defects 2–5 , geometrical defects 6 , and grain ents ͓11͔, nor in experiments that employed samples com- ͓ ͔ boundaries 7 . The most popular model for understanding posed of hard spheres in many different concentration- the transition is Kosterlitz-Thouless-Halperin-Nelson-Young dependent cells ͓12͔. The temperature-sensitive samples ͑ ͒ ͓ ͔ KTHNY theory 2–5 , which predicts two-step melting, employed herein start in the equilibrium crystal phase and from the crystal to a hexatic phase and then from the hexatic reequilibrate rapidly after each tiny temperature jump. There- to a liquid phase. The intermediate hexatic phase has short- fore, they are unlikely to be trapped in metastable glassy range translational and quasi-long-range orientational order, states during melting. Beautiful recent experiments using and the two transitions are beautifully characterized by the magnetic spheres with tunable dipole-dipole interactions creation, binding, and unbinding of topological defects, i.e., ͓13–15͔ share some of these advantages but also differ from dislocations and disclinations, respectively. Experimenters the present experiments in a complementary way as a result have sought out these features across a wide range of mate- of their long-range dipolar interactions. ͓ ͔ rials, including monolayers of molecules and electrons 1 , We measured a variety of sample properties during melt- ͓ ͔ ͓ ͔ liquid crystals 8 , superconductors 9 , diblock copolymers ing, including radial distribution functions, structure factors, ͓ ͔ ͓ ͔ 10 , and colloidal suspensions 11–16 . Some experiments topological defect densities, dynamic Lindemann parameters and simulations have demonstrated substantial agreement ͓13͔, translational and orientational order parameters, and or- with KTHNY theory, but others exhibit deviations and am- der parameter correlation functions in space and time. In this ͓ ͔ biguities possibly due to finite-size effects 1 , the interaction process we discovered that the order parameter susceptibility, ͓ ͔ ͓ ͔ range and form 16 , and out-of-plane fluctuations 17,18 .In i.e., the order parameter fluctuations, proved superior for our view, the collection of evidence clearly points to the finding phase transition points compared to other analyses validity of the KTHNY scenario in 2D systems with long- which typically suffer finite-size and/or finite-time ambigu- ͓ ͔ range interaction potentials 13–15,19 , but the evidence is ities. The susceptibility method has been applied in simula- less convincing in systems with short-range interactions tions ͓17,21͔, but to our knowledge has not been applied in ͓ ͔ 20–24 . Consequently it remains desirable to explore the imaging experiments. Using this method, we clearly resolved phenomena in other model systems, especially those with the intermediate hexatic phase, and we identified a “disloca- short-ranged interactions. tion precursor stage” in the crystal phase that traditional In this paper we examine 2D melting in a colloidal sys- analyses sometimes incorrectly assign to the hexatic phase. tem. The pair potential between particles in this colloidal In addition, the functional form of the susceptibility near the suspension is short ranged and repulsive, and the sphere di- phase transition points, e.g., divergences or discontinuities, ameter is thermally sensitive, so that temperature tuning can can be used to determine the order of the phase transition. be used to vary the sample volume fraction and drive the melting transition. Temperature-sensitive particles enable us II. EXPERIMENT A. Sample preparation and characterization *Present address: Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. The samples consisted of a monolayer of N-isopropyl †Present address: Department of Physics, Ajou University, Suwon acrylamide ͑NIPA͒ spheres confined between two glass cov- 443-749, Korea. erslips. We synthesized NIPA microgel spheres by free-

radical polymerization ͓25͔ and suspended them in buffer 4 (A) ρ=0.909 (26.9°C) (B) solution ͑pH=4.0, 20 mM acetic acid͒. The NIPA polymer ° becomes less hydrophilic at high temperature ͓26͔, and there- 3 30 C ° T] 24 C

fore the sphere diameter decreases with increasing tempera- B

ture as water moves out of the microgel particle. Dynamic [k

) 2 r

light scattering measurements found the NIPA sphere hydro- ( dynamic diameter to vary linearly from 950 nm at 20 °C to u 740 nm at 30 °C and showed that sphere polydispersity was 1 ↓ less than 3%. Such small polydispersity should not affect the ͓ ͔ nature of the melting process 27 . The cleaned glass sur- 0 ↑ faces were coated with a layer of 100-nm-diameter NIPA 0.5 1r[µm] 1.5 spheres to prevent particle sticking. A simple geometrical ρ=0.890 (27.5°C) (C) ρ=0.858 (27.9°C) (D) calculation showed that 100 nm close packed spheres on the surface give rise to ϳ3 nm surface roughness for the 800- nm-diameter spheres. This surface roughness is negligible compared to sphere polydispersity and wall separation fluctuations. In addition, our observations of the large sphere motions at lower concentrations did not find evidence of preferential spatial locations, i.e., of significant surface potentials. The dense monolayer of 800 nm spheres formed crystal domains within the sample cell of typical size ϳ͑40 ␮m͒2, ͑ ͒ ͑ ͒ ͑ ͒ corresponding to ϳ3000 particles. Measurements were car- FIG. 1. a Pair potential u r of NIPA spheres at 24 squares ͑ ͒ ried out on a ϳ͑20 ␮m͒2 central area away from the grain and 30 °C circles . Arrows indicate corresponding hydrodynamic ͑ ͒ ͑ ͒ boundaries. In practice we found that grain boundaries af- radius measured by dynamic light scattering. b – d : Typical 10 s fected only a few neighboring lines of particles, and that particle trajectories in the crystal, hexatic, and liquid phases, respectively. melting started from both inside crystal domains and at grain boundaries. This behavior differs qualitatively from grain- boundary melting in 3D ͓25͔ and edge melting in 2D ͓28͔, temperature using the method described in Ref. ͓31͔. From wherein melting starts from grain boundaries or edges and g͑r͒, we applied liquid structure theory to extract ͓32͔ the then propagates into the crystal. Our observations suggest pair potentials u͑r͒ shown in Fig. 1. Note that the potentials that interfacial energies for liquid nucleation from anywhere are short ranged and repulsive. The effective particle diam- ϳ within the crystal are similar to those near grain boundaries. eter at 1kBT is 10% smaller than the hydrodynamic diam- It appears that melting starts nearly simultaneously through- eter measured by dynamic light scattering. Herein we use the out the crystal. hydrodynamic diameter ␴ for defining the areal density ␳ 2 The sample was heated very slowly in 0.1 °C steps for a =n␲␴ /4, where n is the areal number density. few minutes, and the measurements were taken after the temperature stabilized. We did not observe convection, even in B. Order parameter correlations in space and time samples with smaller particles at much lower volume frac- Figures 1͑b͒–1͑d͒ show typical particle trajectories in the tions. The sample cell was approximately 1 ␮m thick, much three phases. In KTHNY theory, the traditional way to dis- smaller than the 1-mm-thick glass slide on which it was tinguish phases derives from the shapes of the order param- mounted. The temperature difference across the slide was eter correlation functions. For our analysis we first labeled ͑ ␺ ␺ ͒ ␺ about 5 °C, so the gradient across the sample was very small each particle by xj ,yj ,t, 6j , Tj . Here t is time, 6j ␪ ͑Ͻ ͒ ͚͑nn 6i jk͒/N 0.01 °C . Furthermore, the Reynolds number of water at = k=1e is the orientational order parameter, and ␮ ␺ iG·rj this 1 m scale is small, making convection unlikely. Be- Tj=e is the translational order parameter for particle j at ͑ ͒ ␪ fore starting the measurements we cycled the samples once position rj = xj ,yj . jk is the angle of the bond between or twice above the melting point in order to relieve any pos- particle j and its neighbor k. N is the number of nearest sible shear stresses. neighbors ͑NNs͒. G is a primary reciprocal lattice vector Particle motions were observed by microscope and re- determined from the peak of the 2D structure factor s͑k͒ at corded to videotape using a charge-coupled device camera each temperature. In the liquid phase s͑k͒ has no angular ␺ operating at 30 frames/s. The particle positions in each frame peak. Therefore we use G of the crystal phase to compute T were obtained from standard image analysis algorithms ͓29͔. in the liquid phase; this approach has been used previously The temperature control ͑Bioptechs͒ on the microscope had ͓18,33͔. The assignment of G is not always easy in the crys- ␺ slightly better than 0.1 °C resolution. We increased the tem- tal or hexatic phase. To this end we maximized T at each perature from 26.5 to 28.5 °C in 0.2 °C steps and recorded 5 temperature ͑including the liquid phase͒ by iteratively vary- min of video at each temperature after sample equilibration. ing G around an initial estimate derived from s͑k͒; the re- Particle interactions were directly quantified by measuring sultant G was assumed to be optimal for that particular the particle radial distribution function g͑r͒ in a dilute ͑i.e., sample temperature and was used in subsequent calculations areal density ␳ϳ10%͒ monolayer of spheres in the same of order parameter correlation functions and the translational sample cell. We corrected for image artifacts ͓30͔ at each susceptibility.

041406-2 MELTING OF TWO-DIMENSIONAL TUNABLE-DIAMETER … PHYSICAL REVIEW E 77, 041406 ͑2008͒

1 1 (A) (C)

-1/4 r ° (t) (r) 26.5 C -1/8 6 6 ° g g 26.7 C } crystal t ° 0.1 26.9 C 27.1°C 27.3°C } hexatic 27.5°C ° 27.7 C 0.1 27.9°C 28.1°C } liquid 28.3°C 0.01 28.5°C 110r[µm] 0.1 1t [s] 10 100 (B) (D) 0.9 m]

10 (t) 6 [µ g 6

ξ 1 0.8 0.75 0.8ρ 0.85 0.9 0 30 t [s] 60 ͑ ͒͑ ͒ ͑ ͒ FIG. 2. Color online a Orientational correlation functions g6 r . Minima in the oscillations are associated with positions in the lattice /␰ / ͑ ͒ −r 6 −1 4 that are not favored by particles. The five dashed curves are fits of g6 r to e . r is the KTHNY prediction at the hexatic-liquid ͑ ͒ ␰ ͑ ͒ transition point. b Circles are the orientational correlation lengths 6 obtained from the fits in a . The solid curve is a fit to the KTHNY /ͱ␳ ␳ ␰ ͑␳͒ϰ −b␰ i− ␳ prediction 6 e with b␰ =0.566 and i =0.894. These fit values, however, are prone to systematic error as a result of finite-size ͓ ͔ ͑ ͒ ͑ ͒ −1/8 ͑ ͒ effects 34 . c Orientational correlation function g6 t in time. t is the KTHNY prediction at hexatic-liquid transition point. d Expanded version of ͑c͒ that more clearly exhibits the transition from long-range to quasi-long-range order. The 11 temperatures correspond to the 11 densities in Fig. 7.

␺ ␺ Correlations of 6 and T in space and time are readily Despite substantial agreement with the KTHNY model, constructed, yielding four correlation functions two major ambiguities arise in the traditional correlation ͑ ͒ function analyses: 1 The power law decay of g6 can reflect crystal-liquid coexistence rather than the hexatic phase, and ء g␣͑r = ͉r − r ͉͒ = ͗␺␣ ͑r ͒␺␣ ͑r ͒͘, ͑1a͒ i j i i j j ͑2͒ finite-size and finite-time effects induce ambiguities in the correlation function curve shapes near transition points. For example, the T=27.7 °C curve in Fig. 2͑c͒ appears to ء g␣͑t͒ = ͗␺␣ ͑␶͒␺␣ ͑␶ + t͒͘. ͑1b͒ i i decay algebraically over the finite measured time scale, but could decay exponentially at longer times. Since the curve ␣ ͑ ͒ ͑ ͒ / where =6,T. Note that gT r and g6 r are two-body quan- appears below the theoretical t−1 8 transition curve, we ͑per- ͑ ͒ ͑ ͒ tities, and gT t and g6 t are one-body quantities. haps reasonably͒ assigned the system to the liquid phase. ͑ ͒ ͑ ͒ From the g6 r shown in Fig. 2 a , we can semiquantita- tively distinguish three regimes corresponding to crystal, 1

hexatic, and liquid phases as predicted by KTHNY theory: o ͑ ͒ϳ ͑ ͒ 26.5 C g6 r const long-range orientational order for 26.5– o ␩ 26.7 C ͑ ͒ϳ − 6r ͑ ͒ o 26.9 °C, g6 r r quasi-long-range order for 27.1– /␰ 26.9 C ͑ ͒ϳ −r 6 ͑ ͒ o 27.5 °C, and g6 r e short-range order for 27.7– 27.1 C

) o t

( 27.3 C 28.5 °C. These three regions are more clearly resolved over o T 27.5 C three decades of dynamic range in Figs. 2͑c͒ and 2͑d͒, which g o 27.7 C ͑ ͒ ͑ ͒ o plot the dynamic quantity g6 t . Comparing Figs. 2 a and 27.9 C o 2͑c͒, we conﬁrm the KTHNY predictions ͓5͔ that the power 28.1 C o ͑ ͒ ͑ ͒ 28.3 C law decay of g6 t is two times slower than that of g6 r , and o ␩ ␩ / ͑ ͒ 28.5 C 2 6t = 6r =1 4 at the hexatic-liquid transition point. gT t ͑ ͒ and gT r yielded consistent results. For example, Fig. 3 ͑ ͒ϳ −␩ ͑ ͒ Ͻ ͑ ͒ shows that gT t t crystal for T 27 °C and gT t ϳ −t/␶ ͑ ͒ Ͼ e hexatic and liquid for T 27 °C. The KTHNY pre- 0.1 ␩ / ͓ ͔ 0.1 1 10 diction that Tr=1 3 5 at the crystal-hexatic transition t [s] ͑ ͒ ͑ ͒ point was also conﬁrmed. The oscillations in g6 r and gT r correspond to the oscillations of the radial distribution func- FIG. 3. ͑Color online͒ Translational correlation functions gT͑t͒ tion g͑r͒ in Fig. 4. in time.

041406-3 HAN et al. PHYSICAL REVIEW E 77, 041406 ͑2008͒

26.7ºC 27.1ºC 27.5ºC η g(r) ~ r- 45 ξ g(r) ~ e-r/ 26.7ºC

40 26.9ºC 27.7ºC 27.9ºC 28.3ºC 35 27.1ºC 30

27.3ºC 25 FIG. 5. 2D structure factors s͑k͒ at different temperatures. 27.5ºC 20 hexatic phase and simple Lorentzian for the crystal phase ͓12,35͔. However, the proﬁles become quite similar near the crystal-hexatic transition point, and we were unable to accu- 27.7ºC rately determine the transition point from ﬁts to s͑k͒. 15

27.9ºC D. Dynamic Lindemann parameter 10 28.1ºC Another function of interest is the Lindemann parameter, a traditional criterion of melting. For 2D melting, however, the Lindemann parameter diverges slowly even in the crystal 5 28.3ºC phase due to strong long-wavelength fluctuations in 2D. We 28.5ºC calculated the dynamic Lindemann parameter L ͓14͔, defined as 0 051015 2 ͓͗⌬ ͑ ͒ ⌬ ͑ ͔͒2͘ µ ͓͗⌬r ͑t͔͒ ͘ ui t − uj t r[ m] L2 = rel = , ͑3͒ 2a2 2a2 FIG. 4. ͑Color online͒ Radial distribution function g͑r͒ at dif- ⌬ ferent temperatures. Curves are shifted vertically for clarity. The where rrel is the relative neighbor-neighbor displacement, ⌬ thin red and thick blue dashed curves are the power law and the ui is the displacement of particle i, and particles i and j are 2 exponential fits, respectively. nearest neighbors. As shown in Fig. 6, L converges below 27 °C; in this case particles remain near their lattice sites. 2 C. Radial distribution functions and structure factors Divergence of L is found above 27 °C; in this case particles

Other correlation functions, such as the spatial density 10 ͑ ͒ radial distribution function g r in Fig. 4 and the 2D structure o factor s͑k͒ in Fig. 5, also appeared to have ambiguities near 28.5 C 28.3oC phase transition points. The spatial density autocorrelation o function, i.e., the radial distribution function, is defined as 1 28.1 C 2 27.9oC L 1 27.7oC g͑r = ͉r͉͒ = ͗␳͑rЈ + r,t͒␳͑rЈ,t͒͘, ͑2͒ o n2 27.5 C 0.1 27.3oC ␳ ͚N͑t͒␦ ͑ ͒ o where = j=1 (r−rj t ) is the distribution of N particles in 27.1 C the field of view with area A, and n=͗␳͘=͗N͘/A is the areal 26.9oC o density. The angular brackets denote an average over time 0.01 26.7 C and space. The power law fits the data better in the low- 26.5oC temperature crystal phase as shown in Fig. 4. However, both power law and exponential forms fit the data equally well at 0.001 high temperatures. The 2D structure factors in Fig. 5 were 0.1 1 10 100 obtained by Fourier transforming the 2D radial distribution t [sec] functions before azimuthally averaging them to obtain g͑r͒. The expected functional forms of the angular intensity pro- FIG. 6. ͑Color online͒ Square of the dynamic Lindemann pa- file of s͑k͒ from theory are square-root Lorentzian for the rameters at 11 temperatures.

041406-4 MELTING OF TWO-DIMENSIONAL TUNABLE-DIAMETER … PHYSICAL REVIEW E 77, 041406 ͑2008͒

0.01 (A) t=0s (B) t = 2/30 s (C) t = 6/30 s (A) V IV III II I

net disclination

dislocation nn=7 disclination nn=5 disclination

fraction of defects 0 ͑ ͒ 0.04 (B) FIG. 8. Color online Voronoi diagram of the time evolution of

Τ a nonfree dislocation pair at 27.1 °C. Dark blue and light red rep-

0.02 resent particles with ﬁve and seven nearest neighbors, respectively.

͑a͒, ͑b͒, and ͑c͒ all yield zero Burgers vector as shown by the closed hexagonal loop. Dislocations can rapidly form and annihilate in 0

6 pairs, if they are in the same lattice line.

χ (C)

0.02 ͒ ͑ ͒

“free” 6-mer 5–7-5–7-5–7 . In fact, Fig. 7 a very likely

0 ␳ 0.75 0.8 ρ 0.85 ρ ρ 0.9 overestimates m because sufficient numbers of nonfree dis- i m locations are needed before the dislocation chemical poten- FIG. 7. ͑Color online͒͑a͒ Thick dashed curve: N=5 disclination tial reaches zero and free dislocations are produced. Conse- density. Thin dashed curve: N=7 disclination density. Diamonds: quently, a dislocation precursor stage in the crystal phase net disclination density; Circles: dislocation density fitted by might be expected. /͑␳ ␳͒0.36963 We also observed more N=8,9 than N=3,4 defects. This e−2bm m − ͓5͔. ͑b͒ Translational and ͑c͒ orientational suscep- imbalance compensated for some of the density difference tibilities. Dashed curves: ␹L derived from subbox sizes L =5,10,20 ␮m from top to bottom. Symbols: ␹ϱ extrapolated from between five and seven NNs. In fact, a small imbalance in dashed curves. The solid curve in ͑c͒ is a fit to the KTHNY predic- five and seven NN disclinations might be expected. The con- /ͱ␳ ␳ ͓ ͔ ␹ ͑␳͒ϰ −b␹ i− ␳ centration equality holds only in perfect crystals with peri- tion 23 6 e with i =0.901, b␹ =1.14. Vertical solid lines partition crystal ͑regions I and II͒, hexatic ͑region III͒, and odic boundary conditions and neglecting N=3,4,8,... de- liquid ͑regions IV and V͒ phases as determined from susceptibilities fects ͓5͔. Our samples have free boundary conditions. in ͑b͒ and ͑c͒. Region II is a dislocation precursor stage of the Periodic boundary conditions create an artificial constraint crystal with dislocations. Region IV is a prefreezing stage ͓36͔ of that forces vacancies and interstitials to be created in pairs the liquid with ordered patches. rather than diffusing in from the surface ͓5͔. For similar rea- sons, the densities for five and seven NNs are different in our can more readily exchange positions with their neighbors via experiment, but are usually observed to be the same in simu- ͑ ͓ ͔͒ the gliding and climbing of dislocations ͓5͔. This transition at lations e.g., 18 . Furthermore, any deviation from the strict ͓ ͔ 27 °C is consistent with our direct measurement of disloca- monolayer limit can produce a concentration asymmetry 5 . tion densities ͑see discussion below͒ in Fig. 7͑a͒. Besides the static properties noted above, we observed some interesting defect dynamics. For example, dislocations often dissociated from larger defect clusters ͑e.g., 6-mer 5–7- E. Defect densities and dynamics 5–7-5–7͒ rather than from isolated pairs of dislocations ͑the Defect densities are helpful for distinguishing among dif- 5–7-5–7 quartet͒, perhaps because the energy change for ferent phases. Particles with N6 are considered to be de- such disassociation is small. fects. KTHNY theory suggests that the creation of free dis- ͑ ͒ locations isolated 5-7 pairs drives the system from crystal F. Susceptibilities to hexatic phase, and the creation of free disclinations ͑iso- lated N=5 or 7 defects͒ drives the transition from the hexatic In order to avoid the ambiguities outlined above and de- ␳ to the liquid phase. We measured defect concentrations as a termine the true m, we explored the utility of another function of temperature. Figure 7͑a͒ shows that dislocations method for finding phase transition points. This method is Ͼ ͑ ␳ ͒ based on the order parameter fluctuations and is character- start to appear for T 27 °C i.e., m =0.905 , and disclina- Ͼ ͑ ␳ ͒ ized by the order parameter susceptibility tions start to appear for T 27.7 °C i.e., i =0.875 . Al- though defect density measurements are less sensitive to 2 2 2 ␹␣ = L ͉͑͗␺␣͉͘ − ͉͗␺␣͉͘ ͒. ͑4͒ finite-size effects than the correlation functions ͓34͔, the as- L ␳ ␣ ␺ ͚͑N ␺ ͒/ signment of a melting volume fraction m based on defect Here L is the system size, =6,T, and ␣ = i=1 ␣i N is density ͓Fig. 7͑a͔͒ is somewhat problematic. Problems can the total order parameter averaged over all N particles in the ϫ ␹ arise because ͑1͒ the data ͓Fig. 7͑a͔͒ inevitably include dis- L L box. For example, T measures the response of the locations that are not completely “free,” e.g., the dislocation translational order parameter to sinusoidal density fluctua- pairs in Fig. 8͑a͒, that are nearly adjacent to one another, tions with periodicity characterized by G. ␹ is a measure of point in opposite directions, and thus give zero Burgers vec- the fluctuations of the order parameter in 9000 frames. To ␹ tor for a large Burgers circuit, and ͑2͒ the data ͓Fig. 7͑a͔͒ are ameliorate finite-size effects, we calculated L in different susceptible to other systematic errors, e.g., miscounting large size subboxes within the sample ͓dashed curves in Figs. 7͑b͒ defect clusters as equivalent to a free dislocation ͑e.g., a and 7͑c͔͒ and then extrapolated to ␹ϱ, thus attaining the ther-

041406-5 HAN et al. PHYSICAL REVIEW E 77, 041406 ͑2008͒

␹ ͓ ͔ modynamic limit. The L of small subboxes were noisy due liquid 36 because it has visible ordered patches. The non- ␹ ␺ ͑ ͒ to statistics; thus before calculating L, we randomized each zero 6, the splitting of the second peak in g r , and the ͑ ͒ ͑ ␮ ͒2 ͑ ͒ particle’s xj ,yj position within the 20 m box while hexagonal shape of s k in region IV also are indicative of ͑ ␺ ␺ ͒ leaving its t, j6 , jT untouched. Such position randomiza- the presence of ordered patches. Such ordered patches have ␹ ␹ ͓ ͔ tion did not affect L in the largest box, but smoothed L in been observed in simulations 38 , but we are not aware of subboxes, averaging over spatial fluctuations while preserv- reports of a prefreezing stage in 2D melting experiments, ing time fluctuations. Without such a spatial average, the perhaps because it is well known that ordered clusters often dashed curves in Figs. 7͑b͒ and 7͑c͒ became noisier, but the exist in dense fluids. Region V is the liquid phase. same transition points ͑diverging points͒ were resolved. The experimental noise is estimated as the standard deviation of ␺ ͗␺ ͘ H. Phase transition order T in different time frames with respect to the mean T . ͑ ͒ The error bars in Fig. 7 b are estimated from deviations of The order of the phase transition can, in principle, be G ␺ that gave the same noise as found for T. deduced from the shape of the susceptibility curves. If the ␹ ϱ and ␹ ϱ in The sharp divergence or discontinuity of T 6 curve on the left of the diverging point and the curve on the Figs. 7͑b͒ and 7͑c͒ clearly indicate the two transitions of the right of the diverging point have the same asymptotic ␳, then melting process. Although the magnitude of ␹ suffered from the transition is second order; otherwise, it is first order ͓39͔. size effects, the divergence point of ␹ was robust to box size. The curve shape in Figs. 7͑b͒ and 7͑c͒ are consistent with Thus the susceptibility method avoided finite-size ambigu- ␹ ␹ second-order transitions. However, the T curve shape is sen- ities. The divergence of also avoids ambiguities arising sitive to the choice of G even though the divergence point is from the similar functional forms of other measures ͑e.g., ͒ quite robust. For example, using G of the crystal for all correlation functions near transition points. Theoretically we temperatures yielded a curve shape that appeared more like a ␹ to have better statistics than the expect the divergence of first-order transition. correlation function shape, because ␹ is essentially an inte- For the liquid-hexatic transition in Fig. 7͑c͒, when we gral of the correlation function. We also observed that the fitted the left part ͑liquid regime͒ of the curve with the divergence points of the susceptibilities were robust to small KTHNY prediction, we obtained an unreasonably high uncertainties in G, although the exact magnitude of ␹ was T asymptotic transition density ␳ =0.901 ͓21͔. This discrep- G i somewhat sensitive to . ancy suggests that the hexatic-liquid transition may be more first-order-like. In addition, the continuous phase transitions must satisfy universality relations, while first-order transi- G. Five regimes ͑ ͒ tions need not. Our b6␰ =0.566, from Fig. 2 b and b6␹ Five regimes are marked off in Fig. 7 based on the various =1.14, from Fig. 7͑c͒ do not completely satisfy the univer- ͓ ͔ ͑ ␩ ͒ ␩ / analyses we have carried out. Region I is crystal with few sality 23 b6␹ = 2− 6 b6␰ where 6 =1 4. This failure could dislocations ͓Fig. 7͑a͔͒, convergent dynamic Lindemann pa- be viewed as further evidence of a first-order transition; how- ͑ ͒ rameters over the measured time scales Fig. 6 , constant ever, when we forced b ␹ and b ␰ to satisfy the universality ͑ ͒ ͑ ͒͑ ͒ ͑ ͒͑ ͒ 6 6 g6 r and g6 t Fig. 2 , and algebraic decay of gT t Fig. 3 . relation, they still gave somewhat reasonable ͑albeit worse͒ We take region II to be a dislocation precursor stage in the fitting curves because other fitting parameters were adjust- crystal because dislocations have started to appear, but their able too. For example, the five data points in Fig. 2͑b͒ can be density is not high enough for the system to reach the hexatic fitted well by the other two free parameters when b6␰ is fixed. phase, wherein the chemical potential of dislocation reaches In total, the evidence leans slightly to favoring a first-order zero. In other words, the observed dislocations in region II liquid-hexatic transition, but is not sufficient to unambigu- are not free. This gas of nonfree dislocations causes a soft- ously exclude a second-order transition. Future work with ening of the crystal, an effect which has been observed in the finer control of the approach to the phase transition should crystal phase ͓37͔. The dynamic Lindemann parameter is di- enable us to pin down the order of the two transitions more vergent in region II, a direct consequence of the nonzero precisely. dislocation density, which permits particles near dislocations to diffuse out of their cages via the gliding and climbing ͓5͔ ͑ ͒ of dislocations. The correlation function g6 t has finite-size III. CONCLUSIONS ambiguity in region II. For example, the T=27.1 °C curve in Fig. 2͑d͒ appears to have lost orientational order over the In summary, we used the divergence of susceptibilities to measured time scale, but could become constant at longer determine the phase transition points of a 2D colloidal sus- times. If correlation measurements are accurate, then part or pension during the melting process. This approach avoided all of the precursor stage II can be correctly assigned to be in ambiguities from finite-size effects and the divergence points the crystal phase; see Ref. ͓37͔. Interestingly, Li has reana- were robust. We clearly observed the hexatic phase in a sys- lyzed the simulation data in Ref. ͓18͔ with susceptibilities tem of particles interacting via short-range soft-repulsion po- and has also observed the dislocation precursor stage ͓38͔. tentials. Five regimes were assigned to the phase diagram in Region III is the hexatic phase as determined from the ␹ Fig. 7. A number of KTHNY predictions were quantitatively measurements and other analyses. In region IV, disclinations confirmed, especially near the hexatic-liquid transition, but ͓ ͑ ͔͒ ͑ ͒ ͑ ͒ start to appear Fig. 7 a , and g6 r and g6 t decay expo- the order of the two phase transitions was not unambiguously nentially ͑Fig. 2͒. We take region IV to be a prefreezing resolved due to our limited temperature resolution.

041406-6 MELTING OF TWO-DIMENSIONAL TUNABLE-DIAMETER … PHYSICAL REVIEW E 77, 041406 ͑2008͒

ACKNOWLEDGMENTS sions. This work was supported by the MRSEC Grant No. DMR-0520020, NSF Grant No. DMR-0505048, and We thank Dongxu Li, Randy Kamien, David Nelson, Tom Korea Research Foundation Grant No. KRF-2005-214- Lubensky, Noel Clark, and Kirill Korolev for useful discus- C00200.

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